Probalign is a sequence alignment tool that calculates a maximum expected accuracy alignment using partition function posterior probabilities. Base pair probabilities are estimated using an estimate similar to the Boltzmann distribution. The partition function is calculated using a dynamic programming approach.
Algorithm The following describes the algorithm used by probalign to determine the base pair probabilities.
Alignment score To score an alignment of two sequences two things are needed:
a similarity function σ ( x , y ) {\displaystyle \sigma (x,y)} (e.g. PAM, BLOSUM,...) affine gap penalty: g ( k ) = α + β k {\displaystyle g(k)=\alpha +\beta k}
The score S ( a ) {\displaystyle S(a)} of an alignment a is defined as:
S ( a ) = ∑ x i − y j ∈ a σ ( x i , y j ) + gap cost {\displaystyle S(a)=\sum _{x_{i}-y_{j}\in a}\sigma (x_{i},y_{j})+{\text{gap cost}}}
Now the boltzmann weighted score of an alignment a is:
e S ( a ) T = e ∑ x i − y j ∈ a σ ( x i , y j ) + gap cost T = ( ∏ x i − y i ∈ a e σ ( x i , y j ) T ) ⋅ e g a p c o s t T {\displaystyle e^{\frac {S(a)}{T}}=e^{\frac {\sum _{x_{i}-y_{j}\in a}\sigma (x_{i},y_{j})+{\text{gap cost}}}{T}}=\left(\prod _{x_{i}-y_{i}\in a}e^{\frac {\sigma (x_{i},y_{j})}{T}}\right)\cdot e^{\frac {gapcost}{T}}} Where T {\displaystyle T} is a scaling factor. The probability of an alignment assuming boltzmann distribution is given by
P r [ a | x , y ] = e S ( a ) T Z {\displaystyle Pr[a|x,y]={\frac {e^{\frac {S(a)}{T}}}{Z}}}
Where Z {\displaystyle Z} is the partition function, i.e. the sum of the boltzmann weights of all alignments.
Dynamic programming Let Z i , j {\displaystyle Z_{i,j}} denote the partition function of the prefixes x 0 , x 1 , . . . , x i {\displaystyle x_{0},x_{1},...,x_{i}} and y 0 , y 1 , . . . , y j {\displaystyle y_{0},y_{1},...,y_{j}} . Three different cases are considered:
Z i , j M : {\displaystyle Z_{i,j}^{M}:} the partition function of all alignments of the two prefixes that end in a match.
… excerpt ends here. Continue reading the full article.
